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**Characterizations of inner product spaces.**
*(English)*
Zbl 0617.46030

Operator Theory: Advances and Applications, Vol. 20. Basel - Boston - Stuttgart: BirkhĂ¤user Verlag. VII, 200 p. DM 72.00 (1986).

This is an excellent, selfcontained, up-to-date, exhaustive survey on various isometric characterizations of inner product spaces. In Chapter 2 on 2-dimensional characterizations the author discusses among others James and Birkhoff orthogonality characterizations as well as norm derivatives and best approximation characterizations.

In Chapter 3 on 3-dimensional characterizations the characterizations included range from classical Kakutani’s result, through metric projections characterizations to Chebyshev centers and proximinal maximal subspaces, to name just a few topics covered in the book.

The book is well written, well organized. It bears a seal of expertize of the author who is a renowned specialist in the theory of normed spaces.

The book will certainly be a useful source for researchers both in this area as well as in many related areas of functional analysis, geometry and approximation theory.

In Chapter 3 on 3-dimensional characterizations the characterizations included range from classical Kakutani’s result, through metric projections characterizations to Chebyshev centers and proximinal maximal subspaces, to name just a few topics covered in the book.

The book is well written, well organized. It bears a seal of expertize of the author who is a renowned specialist in the theory of normed spaces.

The book will certainly be a useful source for researchers both in this area as well as in many related areas of functional analysis, geometry and approximation theory.

Reviewer: V.Zizler

### MSC:

46C05 | Hilbert and pre-Hilbert spaces: geometry and topology (including spaces with semidefinite inner product) |

46B20 | Geometry and structure of normed linear spaces |

46-02 | Research exposition (monographs, survey articles) pertaining to functional analysis |